B Basic integral

1. Oct 30, 2016

Stephanus

Dear PF Forum,
I'd like to study integral.
But I just realize that I'm lack of basic integral.
Let y=x2
Here is the graph.

So,
1. dx is the distance between the red vertical lines? But it's very, very, very small distance.
2. f(x) * dx is the yellow area?
3. $\int_0^2 x^2\,dx$ is the green (plus yellow) area?
4. And one more thing. The blue line function can be derived from the derivative of f(x)?
For example the derivative of x2 is 2x. So a line that crosses (x,f(x)) where x=1 is 2.
The blue line function is y = 2x.
If it crosses (x,f(x)) where x = 2, then the blue line function is y = 4x?
Thank you very much

2. Oct 30, 2016

Staff: Mentor

We usually call this $\Delta x$.
If $x_i$ is a number in the subinterval at the base of the yellow region, then $f(x_i)\Delta x$ is the area of a rectangle that is approximately equal to the area of the yellow region. The smaller the value of $\Delta x$, the better the approximation is. Here we are approximating a roughly trapezoidal shape by one with a rectangular shape.
Yes
The blue line doesn't "cross" the curve -- it touches it at the point of tangency.
No. The slope of the tangent line (what you're calling the blue line) is 4, but if you draw this line you'll see that the tangent line intersects the y-axis below the origin. The line whose slope is 4, that is tangent to the graph of y = x2 at (2, 4), has this equation: y = 4x - 4.

3. Oct 30, 2016

FactChecker

EDIT: Sorry. I took a break before submitting and this is mostly a repeat of @Mark44's answer above.
Yes.
Yes. But there is a small range of x values in that short dx, so you pick one, say x0, in that range and use f(x0)) * dx as the approximation of the yellow area. As dx gets smaller, the approximation gets better.
Yes.
The derivative f(x0) only gives you the slope of the line tangent at x0. So figuring out the constant term of the line is still required.
Except for the constant. In general, the equation of the tangent line is f'(x0) * (x-x0) + f(x0)

4. Oct 30, 2016

Stephanus

Yes, I think it's just semantic here. I usually see f(u) du or F(v) dv.
But delta is more appropriate I think. The difference, right?
Thanks.
Trapezoidal shape looks very similar to rectangle. Thanks.
Thanks
Yes, I should have drawn it more clearly. What I was going to draw is that the blue line did touch the curve. It touches not crosses. Thanks
Yes, I should have drawn it below the origin. Thanks. So it is slope? In high school we call it gradient.

THanks.

5. Oct 30, 2016

Stephanus

Thanks @FactChecker . I suspect that all the answer should be yes. But I need confirmation. I'm going to read some calculus source. Thanks.

6. Oct 30, 2016

Staff: Mentor

Here the meanings are the same, but the term "gradient" is also used in functions of two or more variables, and in that context, the gradient of f (denoted $\nabla f$) is a vector that consists of the partial derivatives of f. For example, for a function f(x, y, z), $\nabla f$ is the vector $<\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}>$ .

7. Oct 30, 2016

FactChecker

I would only add that the gradient vector points in the direction of fastest increase of the function and its magnitude is the slope in that direction.